So I can say I have energy in space, and I want to look at the rate of change of energy with respect to time. Then there is some power of energy, sound energy coming in. And the rate of change of total energy, kinetic plus potential inside of room enclosed by the impedance discontinuity in space has to be balanced by these two variables, power in and power out. Again, if you think that's an analogy with your bank account, the increase of money in your bank account per one day would be balanced by how much money coming in, and how much money coming out. If the money coming out is larger than coming in, then obviously, the rate of change of your bank account is negative. Okay. Also, I can say this one has two parts. One is direct energy and one is reverberant. In other words, when I say something over here like, and what you can hear is something directly coming from me, and it's something which already hit the wall and back to you. So distance from here to there, there are many, many different distances within the certain time constant, t, okay? And I can write the following things alsa about this. Power in, it has one component. The power in is power in due to direct, and power out, I have two components, which is due to direct and pie out due to reverberant. Yes. This is a very obvious. This is power in direct from the speaker or source, and this is what would be out not reflecting anywhere. When I generate the sound, there the energy would come out due to the impedance mismatch over there. Okay, that is the energy directly coming out from the room. That is this part. This part for example, there will be some reflection due to the impedance mismatch, and then reflected wave again, meet this wall, and then coming out, okay? Therefore, I can say from this equation d dt, EdV, reverberant especially has to be equal to minus pie out reverberant. All right? In other words, we only would like to consider the energy increase or decrease inside of the room. Not direct field, only reverberant field associated with the pie power out reverberant. Okay, this is very direct, okay? And then, if the E reverberant in average sense would be constant with respect to space, not the time, it will change with respect to time. Because for example, it will change with respect to time, but what I'm saying is a constant with respect to space. And I denote this is E, I would use this notation and that is constant with respect to space. Then Sabine found that pie out reverberant is proportional to the size of room and energy overall. That is great observation. Physical meaning of that is how much energy is coming out. It depends on or proportionate to the size of room. If the size of room is large, then if it is reverberant or diffused, the total energy inside of the room would be very large. Because we are feeling the energy continuously, like you are listening orchestra in the room, then the energy in the room has to be filled out all the way. So, it is proportional to the size of room. Large size of the room, large energy out. And also he found that for the reverberant room case, the pie power out is also proportionate to the energy being energy density of the room. If you have a very large energy density in the room, then you've got more power out. So that is one findings. So, we can write therefore, from this relation, we can find the reverberant dt is proportional to because he found this. We can write this is proportional to V and E reverberant. Okay. And the over there, we have to integrate. And if this is constant with respect to space then the integration give me the volume. Therefore, I can write d E reverberant divided by d T is proportional to energy reverberant. This is very interesting. And then, he tried to find a way to make this a proportional to the equality, and that is obviously d E reverberant d T is proportional to E reverberant. And to be equal, something has to come over here, has to be the divider of one over Time. Because this is a DDT and that is energy, therefore that has to be one over time constant which we do not know yet. Okay that is nice. And this has to be because the energy is coming out, this has to be, this has to be cannot be positive. Energy has to be decay. I mean, energy cannot increase so it has to be minus. Now, this is a very simple differential equation and we do know the solution and E reverberant is equal to initial energy which I denote to be zero, and exponential decay minus T over time. Oh that's very pretty obvious. That means when I hit sound over there, then the energy is exponentially decay like you may hear the sound is decaying very rapidly with the some time constant. So. this is very simple straightforward result. The physical meaning is that, the sound in the room can characterize this train time constant tau. In other words, tau determines room acoustics, or tau determines the quality overall, in other words. So, what is tau? So, if you look at this equation when tau is large than energy, and tau is large then the energy is rapidly decaying or slowly decaying when tau is large. Say tau is 100 then the energy is slowly decaying. When tau is small, the energy will decay very rapidly. And then, Sabine did the experiment again and he found that interestingly, that tau is proportional to the size of room and open area window, As. If size of room is big, then energy is just slowly decaying. Suppose you are shouting in a big auditorium or big gymnasium for example and if you shouted, you will hear sort of echoes and then, you can see that the energy is just slowly decaying. But if you go to the small room like this room, then energy will not decay as slowly as you observed in the big auditorium. And also, if you go to, for example bathroom and if you shout then, As in the bathroom is very, very small, therefore Tao would be very large, therefore energy decay should be very slow as you normally experience in your day life. Also if you enter [inaudible] room then As will be very large then tau is a very small, therefore energy decay as you experienced before will be very rapid. So that makes sense. And also, by doing a lot of experiment, he found that tau because this is the dimension of L square, LQ and that this is the dimension of L square, so this is the dimension of L and that this is the dimension of the time. Therefore, if I divided this value by speed of sound then this will have a dimension of time, an experiment he found that there is a magic number four. Wow, this is a great finding magic number four. So, as a conclusion, we can write to this energy decay equation defined by famous Sabine.
